164] STEADY FLOW IN CONDUCTORS. 329 



lies the surface B\ we may show that in the different conductors 

 the potential is given by the functions 



V = 



v 3 

 (H) 



For since the function <E> satisfies the differential equation 

 that is satisfied by the potential, any v gt which is a linear function 

 of <i>, must also satisfy the same equation. Also at any surface 

 separating the conductors r and r + 1, 



and 



H 

 dn r + dn r+l 



from the definition of the function <I>. At the insulating boun- 

 dary of any conductor 



that is, there is no flow across the boundary. The function v l takes at 

 the surface A the value V A , and the function v n at the surface B 

 the value V B . But these are all the conditions satisfied by the 

 potential function. It remains to show that the function <3> is 

 uniquely determined by the conditions that have been imposed 

 upon it. The problem of finding the function <I> is of the same 

 nature as Dirichlet's problem, differing from it in that while 

 the values of <I> are given over part of the bounding surface^ 

 over the remainder instead of <1> the values of d&/dn are given. 



Suppose that there are two functions <3> both satisfying the 

 conditions of definition. Let them be denoted by <>! and <E> 2 

 Then let us form the integral taken throughout the conductors 

 considered 



, f f 



ii] 



